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Chapter 39: Q.51 (page 1140)

Consider the electron wave function

ψx=cxx≤1nmcxx≤1nm

where x is in nm. a. Determine the normalization constant c.

b. Draw a graph of c1x2 over the interval -5 nm … x … 5 nm. Provide numerical scales on both axes.

c. Draw a graph of 0 c1x2 0 2 over the interval -5 nm … x … 5 nm. Provide numerical scales.

d. If 106 electrons are detected, how many will be in the interval -1.0 nm … x … 1.0 nm?

Short Answer

Expert verified

The wave function of the electronψx=cxx≤1nmcxx≤1nm

Step by step solution

01

The probability independent

∫-∞+∞ψxdx=1∫-∞+∞ψxdx+∫-11ψxdx+∫12ψxdx=1∫-∞+∞cx2dx+∫-11cxdx+∫12cx2dx=1c2223=1

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Most popular questions from this chapter

a. Starting with the expressionΔfΔt≈1for a wave packet, find an expression for the product

ΔEΔtfor a photon.

b. Interpret your expression. What does it tell you?

c. The Bohr model of atomic quantization says that an atom in an excited state can jump to a lower-energy state by emitting a photon. The Bohr model says nothing about how long this process takes. You'll learn in Chapter 41 that the time any particular atom spends in the excited state before emitting a photon is unpredictable, but the average lifetime Δtof many atoms can be determined. You can think of Δtas being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is Δt≈10ns. Consider an atom that emits a photon with a 500nmwavelength as it jumps down from an excited state. What is the uncertainty in the energy of the photon? Give your answer in eV.

d. What is the fractional uncertainty ΔE/Ein the photon's energy?

FIGURE P39.32 shows |ψ(x)|2for the electrons in an experiment.

a. Is the electron wave function normalized? Explain.

b. Draw a graph of ψ(x)over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)

c. What is the probability that an electron will be detected in a 0.0010-cm-wide region atx=0.00cm? At x=0.50cm? At x=0.999cm?

d. If 104electrons are detected, how many are expected to land in the interval -0.30cm≤x≤0.30cm?

A proton is confined within an atomic nucleus of diameter4.0m. Use a one-dimensional model to estimate the smallest range of speeds you might find for a proton in the nucleus.

Soot particles, from incomplete combustion in diesel engines, are typically 15nmin diameter and have a density of 1200kg/m3. FIGURE P39.45 shows soot particles released from rest, in vacuum, just above a thin plate with a 0.50-μm-diameter holeroughly the wavelength of visible light. After passing through the hole, the particles fall distance dand land on a detector. If soot particles were purely classical, they would fall straight down and, ideally, all land in a 0.50-μm-diameter circle. Allowing for some experimental imperfections, any quantum effects would be noticeable if the circle diameter were 2000nm. How far would the particles have to fall to fill a circle of this diameter?

The probability density for finding a particle at position xis

px=a1-xb1-x-1mm≤x<0mm0mm≤x≤1mm

and zero elsewhere
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